A224900 a(n) = n!*((n+1)!)^2.

1, 4, 72, 3456, 345600, 62208000, 18289152000, 8193540096000, 5309413982208000, 4778472583987200000, 5781951826624512000000, 9158611693373227008000000, 18573664514160904372224000000, 47325697182081984340426752000000, 149075946123558250672344268800000000

Offset: 0

Author

Karol A. Penson, Jul 24 2013

Keywords

• nonn
• easy

Comments

2*a(n-1) is the number of elements of the wreath product of S_n and S_3 with cycle partition equal to (3n). - Josaphat Baolahy, Mar 12 2024

Crossrefs

Cf. A172492.

Programs

Mathematica

Table[n!*((n+1)!)^2,{n,0,15}]

Python

from math import factorial
def A224900(n): return factorial(n)**3*(n+1)**2 # Chai Wah Wu, Apr 22 2024

Formula

G.f. of hypergeometric type: Sum_{n>=0} a(n)*z^n/(n!)^3 = (1+z)/(1-z)^3.

Integral representation as n-th moment of a positive function w(x) on a positive half axis (solution of the Stieltjes moment problem), in Maple notation: a(n) = int(x^n*w(x),x=0..infinity), n>=0, where w(x) = MeijerG([[],[]],[[1,1,0]],[]],x), w(0)=1, limit(w(x),x=infinity)=0.

w(x) is monotonically decreasing over (0,infinity).

The Meijer G function above cannot be represented by any other known special function.

This solution of the Stieltjes moment problem is not unique.

Asymptotics: a(n) -> (1/960)*sqrt(2)*Pi^(3/2)*(1920*n^3 + 4320*n^2 + 2940*n + 589)*exp(-3*n)*n^(1/2 + 3*n), for n->oo.

a(n) = A172492(n)*(n+1).

a(n) - n*(n+1)^2*a(n-1) = 0. - R. J. Mathar, Jul 28 2013